Section: Research Program

Sequences

Participants : Alain Denise, Mireille Régnier, Yann Ponty, Jean-Marc Steyaert, Alice Héliou, Daria Iakovishina, Antoine Soulé.

String searching and pattern matching is a classical area in computer science, enhanced by potential applications to genomic sequences. In Cpm/Spire community, a focus is given to general string algorithms and associated data structures with their theoretical complexity. Our group specialized in a formalization based on languages, weighted by a probabilistic model. Team members have a common expertise in enumeration and random generation of combinatorial sequences or structures, that are admissible according to some given constraints. A special attention is paid to the actual computability of formula or the efficiency of structures design, possibly to be reused in external software.

As a whole, motif detection in genomic sequences is a hot subject in computational biology that allows to address some key questions such as chromosome dynamics or annotation. This area is being renewed by high throughput data and assembly issues. New constraints, such as energy conditions, or sequencing errors and amplification bias that are technology dependent, must be introduced in the models. An other aim is to combine statistical sampling with a fragment based approach for decomposing structures, such as the cycle decomposition used within F. Major's group  [66] . In general, in the future, our methods for sampling and sequence data analysis should be extended to take into account such constraints, that are continuously evolving.

Combinatorics of motifs

Participants : Mireille Régnier, Alice Héliou, Daria Iakovishina.

Besides applications [5] of analytic combinatorics to computational biology problems, the team addressed general combinatorial problems on words and fundamental issues on languages and data structures.

Molecular interactions often involve specific motifs. One may cite protein-DNA (cis-regulation), protein-protein (docking), RNA-RNA (miRNA, frameshift, circularisation). Motif detection combines an algorithmic search of potential sites and a significance assessment. Assessment significance requires a quantitative criterium. It is generally accepted that the p-value is a reliable tool that outperforms older criteria such as the z-score. Amib develops a long term research on word combinatorics. In the recent years, a general scheme of derivation of analytic formula for the pvalue under different constraints ($k$-occurrence, first occurrence, overrepresentation in large sequences,...) has been provided. It relies on a representation of word overlaps in a graph [40] . Recursive equations to compute pvalues may be reduced to a traversal of that graph, leading to a linear algorithm. It allows for a derivation of pvalues, decreasing the space and time complexity of the generating function approach or previous probabilistic weighted automata.

In the mean time, continuous sequences of overlapping words, currently named clumps or clusters turn out to be crucial in random words counting. Notably, they play a fundamental role in the Chen-Stein method of compound Poisson approximation. A first characterization was proposed by Nicodème and al. and this work is currently extended.

This research area is widened by new problems arising from de novo genome assembly or re-assembly. For example, unique mappability of short reads strongly depends of the repetition of words. Although the average values for the length have been studied for long under different constraints, their distribution or profile remained unknown until the seminal paper [67] which provides formulae for binary tries. A collaboration has been started with Lob at Ecole Polytechnique to check these formulae on real data, namely Archae genomes (internship of D. Busatto-Gaston). This collaboration has been extended since Lob bought a sequencing machineand a co-advisd thesis (Alice Héliou) on circular RNA characterization hasjust started.

As a third example, one objective is to develop a model of errors, including a statistical model, that takes into account the quality of data for the different sequencing technologies, and their volume. This is the subject of an international collaboration with V. Makeev's lab (IoGene, Moscow) and Magnome project-team. Finally, Next Generation Sequencing open the way to the study of structural variants in the genome, as recently described in [48] . Defining a probabilistic model that takes into account main dependencies -such as the GC content- is a task of D. Iakovishina's thesis, to be defended in 2015, in a collaboration with V. Boeva (Curie Institute).

Random generation

Participants : Alain Denise, Yann Ponty.

Analytical methods may fail when both sequential and structural constraints of sequences are to be modelled or, more generally, when molecular structures such as RNA structures have to be handled. The random generation of combinatorial objects is a natural, alternative, framework to assess the significance of observed phenomena. General and efficient techniques have been developed over the last decades to draw objects uniformly at random from an abstract specification. However, in the context of biological sequences and structures, the uniformity assumption becomes unrealistic, and one has to consider non-uniform distributions in order to derive relevant estimates. Typically, context-free grammars can handle certain kinds of long-range interactions such as base pairings in secondary RNA structures.

In 2005, a new paradigm appeared in the ab initio secondary structure prediction [55] : instead of formulating the problem as a classic optimization, this new approach uses statistical sampling within the space of solutions. Besides giving better, more robust, results, it allows for a fruitful adaptation of tools and algorithms derived in a purely combinatorial setting. Indeed, we have done significant and original progress in this area recently  [68] , [5] , including combinatorial models for structures with pseudoknots. Our aim is to combine this paradigm with a fragment based approach for decomposing structures, such as the cycle decomposition used within F. Major's group  [66] .

Besides, our work on random generation is also applied in a different fields, namely software testing and model-checking, in a continuing collaboration with the Fortesse group at Lri   [53] , [65] .